Solving For Angles A And B Given Trigonometric Conditions

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In the realm of trigonometry, solving for unknown angles given certain conditions is a fundamental skill. This article delves into a specific problem where we are given the values of sine and cosine for the sum and difference of two angles, A and B, and are tasked with finding the values of these angles. This problem not only reinforces our understanding of trigonometric identities but also highlights the importance of considering the given constraints to arrive at a unique solution. Let's break down the problem step by step, leveraging our knowledge of trigonometric functions and their properties to unveil the values of angles A and B. This exploration will serve as a valuable exercise for anyone keen on mastering trigonometry and its applications.

Given that $\sin (A-B)=\frac{1}{2}$ and $\cos (A+B)=\frac{1}{2}$, where $0^{\circ} < A+B < 90^{\circ}$ and $A > B$, we aim to determine the values of angles $A$ and $B$. This problem presents a classic scenario in trigonometry where we utilize the properties of sine and cosine functions along with the given constraints to solve for unknown angles. The conditions $0^{\circ} < A+B < 90^{\circ}$ and $A > B$ are crucial as they help us narrow down the possible solutions and ensure that we arrive at a unique pair of values for $A$ and $B$. Solving this problem involves a combination of trigonometric knowledge, algebraic manipulation, and logical deduction. By carefully analyzing the given equations and constraints, we can systematically unravel the values of the unknown angles. This exercise not only enhances our problem-solving skills in trigonometry but also deepens our understanding of the relationships between angles and trigonometric functions. Understanding these relationships is essential for various applications in fields such as physics, engineering, and navigation.

Before diving into the solution, let's solidify our understanding of some fundamental trigonometric concepts. The sine function, denoted as $\sin(\theta)$, represents the ratio of the length of the side opposite to the angle $\theta$ to the length of the hypotenuse in a right-angled triangle. The cosine function, denoted as $egin{aligned}cos(\theta)\end{aligned}$, represents the ratio of the length of the side adjacent to the angle $\theta$ to the length of the hypotenuse. These functions are periodic, meaning their values repeat after a certain interval, and they exhibit specific properties within different quadrants of the unit circle. For instance, the sine function is positive in the first and second quadrants, while the cosine function is positive in the first and fourth quadrants. Furthermore, we need to recall the values of sine and cosine for some special angles, such as 30Β°, 45Β°, and 60Β°. Specifically, $\sin(30^{\circ}) = \frac{1}{2}$ and $\cos(60^{\circ}) = \frac{1}{2}$. These values will be instrumental in solving our problem. Additionally, understanding the relationships between trigonometric functions of complementary angles is crucial. For example, $\sin(\theta) = \cos(90^{\circ} - \theta)$. By mastering these basic concepts and relationships, we can approach trigonometric problems with confidence and clarity. In the context of our problem, these foundational principles will guide us in interpreting the given equations and deriving the values of angles A and B.

  1. Analyze the Given Equations: We are given two equations: $\sin(A-B) = \frac{1}{2}$ and $\cos(A+B) = \frac{1}{2}$. The first step is to recognize the angles for which sine and cosine have the value of $\frac{1}{2}$. We know that $\sin(30^{\circ}) = \frac{1}{2}$ and $\cos(60^{\circ}) = \frac{1}{2}$. Therefore, we can deduce that $A-B$ must equal 30 degrees and $A+B$ must equal 60 degrees. This initial analysis sets the stage for the subsequent steps, where we will use these deductions to form a system of linear equations. By converting the trigonometric equations into algebraic equations, we simplify the problem and make it easier to solve for the unknown angles. This step highlights the importance of recognizing key trigonometric values and their corresponding angles. Without this knowledge, it would be difficult to proceed with the solution. Furthermore, this step demonstrates the power of linking trigonometric concepts with algebraic techniques to solve complex problems. By strategically applying trigonometric identities and algebraic principles, we can effectively unravel the unknowns in a given problem.
  2. Formulate Equations: From the analysis in step 1, we can establish two equations:

    • Aβˆ’B=30∘A - B = 30^{\circ}

    • A+B=60∘A + B = 60^{\circ}

    This system of linear equations is the cornerstone of our solution. Each equation represents a relationship between the angles $A$ and $B$. By solving this system, we can simultaneously determine the values of both angles. The formation of these equations is a crucial step as it transforms the trigonometric problem into an algebraic one, which is often easier to handle. This transformation underscores the interconnectedness of different branches of mathematics and the power of using one branch to solve problems in another. The equations themselves are simple linear equations, but their derivation from the original trigonometric problem demonstrates the elegance and efficiency of mathematical reasoning. By carefully translating the given information into mathematical expressions, we pave the way for a systematic and accurate solution. The next step will involve employing algebraic techniques to solve this system of equations and find the values of $A$ and $B$.
  3. Solve the System of Equations: We now have a system of two linear equations with two unknowns. There are several methods to solve such a system, such as substitution or elimination. Let's use the elimination method. Adding the two equations:

    (Aβˆ’B)+(A+B)=30∘+60∘(A - B) + (A + B) = 30^{\circ} + 60^{\circ}

    2A=90∘2A = 90^{\circ}

    A=45∘A = 45^{\circ}

    Now, substitute the value of $A$ into either equation to find $B$. Let's use the second equation:

    45∘+B=60∘45^{\circ} + B = 60^{\circ}

    B=60βˆ˜βˆ’45∘B = 60^{\circ} - 45^{\circ}

    B=15∘B = 15^{\circ}

    Thus, we have found the values of $A$ and $B$. The elimination method is a particularly efficient way to solve this system of equations because it allows us to directly eliminate one of the variables, making it easier to solve for the other. The process of adding the equations together is a key step in this method, and it relies on the fundamental principle that adding equal quantities to both sides of an equation preserves the equality. The resulting equation, $2A = 90^{\circ}$, is a simple linear equation that can be easily solved for $A$. Once we have the value of $A$, we can substitute it back into one of the original equations to solve for $B$. This substitution step is a common technique in algebra and is used to reduce the number of unknowns in an equation. By carefully applying these algebraic techniques, we have successfully solved for the values of $A$ and $B$, which are the solutions to our problem.
  4. Verify the Solution: We found that $A = 45^{\circ}$ and $B = 15^{\circ}$. Let's verify these values against the given conditions:

    • A + B = 45^{\circ} + 15^{\circ} = 60^{\circ}$, which satisfies the condition $0^{\circ} < A + B < 90^{\circ}$.

    • A - B = 45^{\circ} - 15^{\circ} = 30^{\circ}$, and $\sin(30^{\circ}) = \frac{1}{2}$.

    • A > B$ is also satisfied since $45^{\circ} > 15^{\circ}$.

    Therefore, our solution is consistent with the given information. Verification is a crucial step in any mathematical problem-solving process. It allows us to catch any errors that may have occurred during the solution process and ensures that our final answer is correct. In this case, we verified our solution by plugging the values of $A$ and $B$ back into the original equations and conditions. This process confirmed that our solution satisfies all the given constraints, giving us confidence in our answer. The act of verifying a solution not only ensures accuracy but also deepens our understanding of the problem and the relationships between the variables. By taking the time to verify our work, we strengthen our problem-solving skills and develop a more rigorous approach to mathematics. This step highlights the importance of attention to detail and a commitment to ensuring the correctness of our results.

Therefore, the values of the angles are $A = 45^{\circ}$ and $B = 15^{\circ}$.

The problem we've solved, while seemingly abstract, has significant practical implications in various fields. Trigonometry, in general, is the backbone of many scientific and engineering disciplines. For instance, in navigation, understanding angles and trigonometric functions is crucial for determining the position and direction of a vessel. In physics, analyzing projectile motion and wave behavior heavily relies on trigonometric principles. Engineers use trigonometry to design structures, calculate forces, and analyze stresses. Even in computer graphics and game development, trigonometric functions are used to create realistic movements and visual effects. Specifically, the type of problem we tackled, involving the sum and difference of angles, often arises in situations where we need to decompose a complex motion or force into simpler components. For example, in robotics, understanding the angles of joints and links is essential for controlling the movement of a robot arm. Similarly, in signal processing, the analysis of waveforms often involves breaking down complex signals into simpler sinusoidal components using trigonometric techniques. The ability to solve trigonometric equations and find unknown angles is therefore a valuable skill for anyone pursuing a career in science, technology, engineering, or mathematics (STEM) fields. By mastering these fundamental concepts, students and professionals can unlock a wide range of opportunities and contribute to advancements in various industries. This problem serves as a reminder that mathematical concepts, even those that appear theoretical, have real-world applications that impact our daily lives.

When solving trigonometric problems like this, there are several common pitfalls that students often encounter. Being aware of these pitfalls and knowing how to avoid them can significantly improve your problem-solving accuracy. One common mistake is forgetting the periodic nature of trigonometric functions. Sine and cosine functions repeat their values every 360 degrees, so there can be multiple angles that satisfy a given trigonometric equation. In our problem, we avoided this pitfall by using the constraint $0^{\circ} < A + B < 90^{\circ}$ to narrow down the possible solutions. Another common mistake is incorrectly applying trigonometric identities. It's crucial to have a solid understanding of these identities and to use them appropriately. For example, confusing the sum and difference identities for sine and cosine can lead to incorrect results. A third pitfall is making algebraic errors when solving the system of equations. Careless mistakes in addition, subtraction, or substitution can throw off the entire solution. To avoid this, it's essential to double-check your work and to use a systematic approach, such as the elimination method, to minimize errors. Finally, failing to verify the solution is a significant pitfall. Even if you've followed all the steps correctly, it's always a good idea to plug your answer back into the original equations and conditions to ensure that it satisfies them. This step can catch any subtle errors that may have slipped through the initial solution process. By being mindful of these common pitfalls and adopting a careful, systematic approach to problem-solving, you can significantly improve your chances of success in trigonometry.

In conclusion, we have successfully determined the values of angles $A$ and $B$ given the conditions $\sin (A-B)=\frac{1}{2}$ and $egin{aligned}cos (A+B)=\frac{1}{2}\end{aligned}$, with the constraints $0^{\circ} < A+B < 90^{\circ}$ and $A > B$. Through a systematic approach involving trigonometric analysis, equation formulation, algebraic manipulation, and solution verification, we arrived at the solution $A = 45^{\circ}$ and $B = 15^{\circ}$. This problem serves as a valuable exercise in applying trigonometric principles and problem-solving techniques. The key to success lies in a solid understanding of trigonometric functions, their properties, and the ability to translate trigonometric equations into algebraic ones. Furthermore, the importance of considering given constraints and verifying the solution cannot be overstated. By mastering these skills, one can confidently tackle a wide range of trigonometric problems and appreciate their applications in various fields. The process of solving this problem not only reinforces mathematical knowledge but also cultivates critical thinking and analytical skills, which are essential for success in any academic or professional endeavor. Trigonometry, as demonstrated by this example, is a powerful tool for understanding and modeling the world around us, and a strong foundation in this subject is invaluable for anyone pursuing a career in STEM fields.